The Adaptive Inverse Control Concept

1.0 INTRODUCTION

Adaptive filtering techniques have been successfully applied to adaptive antenna systems [1-20]; to communications problems such as channel equalization [21-30] and echo cancelation in long-distance telephony [31-39]; to interference canceling [40-46]; to spectral estimation [47-57]; to speech analysis and synthesis [58-60]; and to many other signal processing problems. It is the purpose of this book to show how adaptive filtering algorithms can be used to achieve adaptive control of unknown and possibly time varying systems.

The system to be controlled, usually called the "plant," may be noisy, that is, subject to disturbances, and for the most part it may be unknown in character.¹ The plant and its internal disturbances may be time variable in an unknown way. In some cases, the plant might even be unstable. Adaptive control systems for such plants would be advantageous over fixed systems since the parameters of adaptive systems can be adjusted or tailored to the unknown and varying requirements of the plant to be controlled. Adaptivity finds a natural area of application in the control field [88].

In the past two decades or so, many hundreds of papers have been published on adaptive control systems in the Transactions of the IEEE Control Systems Society, in Automatica, in the IFAC (International Federation for Automatic Control) journals and conference proceedings, and elsewhere. At the same time, a very large number of papers on adaptive signal processing and adaptive array processing have appeared in the Transactions of the IEEE Signal Processing Society, Antennas and Propagation Society, Communications Society, Circuits and Systems Society, Aerospace and Electronics Society, the Proceedings of the IEEE, and elsewhere. Many books have been published on these subjects. The two schools of thought, adaptive controls and adaptive signal processing, have developed almost independently. The control theorists have by and large studied adaptive control using state variable feedback coupled through variable parameter networks to regulate unknown plants and to control their disturbances. The signal processing people have been working on problems that for the most part involve adapting weights of transversal filters by gradient methods and employing the resultant adaptive filters to systems without feedback (except for feedback in the adaptive process itself). The signal processing people have found a great number of practical applications for their work, and so have the adaptive control people.

The goal of this book is not to bridge the gap between these two schools of thought but to attack certain problems in adaptive control from an alternative point of view using the methodology of adaptive signal processing. The result is what we call "adaptive inverse control."

We begin with a discussion of direct modeling (or identifying) the characteristics of the unknown plant using simple adaptive filtering methods. Then we show how similar methods, with some modification but in a different configuration, can be used for inverse modeling (or equalization or deconvolution). Inverse plant models can be used to control plant dynamics. Next we show how both direct and inverse models can be used in the same adaptive process to minimize the effects of plant disturbance. In this development, we assume that the plant is completely controllable and observable, that it can (in a quasistatic sense) be represented in terms of an input-output transfer function (albeit an unknown one), and that the plant is stable (if unstable, someone has previously applied stabilization feedback). The plant may be either minimum-phase or nonminimum-phase.

The basic ideas of adaptive inverse control have been under development at Stanford University over the course of many years. The earliest related work is described in a paper by Widrow on blood pressure regulation [61]. Subsequent work is reported in several papers that were presented at Asilomar conferences [62,63]. A Ph.D. dissertation by Shmuel Schaffer was concerned with model-reference adaptive inverse control [64]. A tutorial on the work is given by Widrow and Stearns [65 ]. The first paper on adaptive inverse control including adaptive plant disturbance canceling was presented by Widrow and Walach in 1983 at the First IFAC Workshop in Control and Signal Processing in San Francisco [66]. The second presentation was by Widrow in 1986 in a keynote talk at the Second IFAC Workshop on Adaptive Systems in Control and Signal Processing, University of Lund, Sweden [67]. There have been almost no other publications on inverse control and disturbance canceling until recently. Several recent publications in the neural network literature have appeared concerning nonlinear adaptive inverse control [95,96,97].

¹ Some prior knowledge of the character of the plant and its internal disturbances will be needed in order to establish proper control. For example, at least a rough idea of the transient response time of the plant would be required in order to model it adaptively. Some idea of how rapidly the plant characteristics change for plants that vary over time would be needed. Some knowledge of the plant disturbance would be useful, such as disturbance power level at the plant output. Detailed knowledge of the plant and its disturbances would not be required however.